Yes you can file a motion with the court for an extension of time to complete the discovery for the purpose of designating an additional expert(s). See Texas Rules of Civ. Proc. 5(b).
The Texas rules of procedure have specific time limits with regard to when expert witness designation and reporting must be completed with respect to the discovery window. See Texas Rule of Civ. Procedure 195.2. Since the defense has extended the discovery period you may have time to add an expert. You can ask the court for a modification of any current discovery deadline effectively expanding the discovery window further. The extension should be limited to as short a window as feasibly possible.
You should also note that when a deadline has passed it is much more difficult to get the court to allow an extension. You will need to provide a good reason for requesting the extension. It also helps if the other party does not object to the extension of time to add an expert.
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